Math 2.0

statement proposition

The exponential function is strictly increasing

Statement

The exponential function is strictly increasing on R\mathbb{R}. That is, x<yx < y implies exp⁡(x)<exp⁡(y)\exp(x) < \exp(y).

Remarks

The derivative is exp⁡\exp itself, and it is positive. Then positive derivative gives the claim. Also exp⁡(x)→∞\exp(x) \to \infty as x→∞x \to \infty and exp⁡(x)→0\exp(x) \to 0 as x→−∞x \to -\infty. So exp⁡\exp is a bijection from R\mathbb{R} onto (0,∞)(0, \infty). Mathlib states it as Real.exp_strictMono.

Explanations

L1 human

The exponential function always grows as xx grows. So it takes each positive value exactly once.

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What this needs

To read this node, read these 48 nodes in this order. The foundations come first, and the list ends with this node.

  1. Limit point of a set definition
  2. Local maximum definition
  3. Sequence definition
  4. Strictly monotone function definition
  5. Limit of a function definition
  6. Local minimum definition
  7. Real sequence definition
  8. Subsequence definition
  9. Continuity at a point definition
  10. Derivative definition
  11. Bounded sequence definition
  12. Geometric sequence definition
  13. Limit of a sequence definition
  14. Monotone sequence definition
  15. Series definition
  16. Continuous function definition
  17. Fermat's theorem on stationary points statement
  18. Sum of a finite geometric series statement
  19. Monotone convergence theorem for sequences statement
  20. Convergent sequence definition
  21. Every sequence has a monotone subsequence statement
  22. Squeeze theorem statement
  23. Cauchy product definition
  24. Partial sum definition
  25. Power series definition
  26. Bolzano–Weierstrass theorem statement
  27. The sequence r^n tends to zero for |r| < 1 example
  28. Radius of convergence definition
  29. Convergent series definition
  30. Boundedness theorem statement
  31. Absolute convergence definition
  32. A series of nonnegative terms converges iff its partial sums are bounded statement
  33. Sum of a series definition
  34. Extreme value theorem statement
  35. Comparison test statement
  36. Geometric series example
  37. Cauchy product of absolutely convergent series statement
  38. Rolle's theorem statement
  39. Ratio test statement
  40. Mean value theorem statement
  41. Ratio formula for the radius of convergence statement
  42. A function with positive derivative is strictly increasing statement
  43. The exponential series has infinite radius of convergence statement
  44. Exponential function definition
  45. Exponential of a sum statement
  46. The exponential function is positive statement
  47. The exponential function is its own derivative statement
  48. The exponential function is strictly increasing statement

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18 nodes use this node, directly or through other nodes.

Sources

  1. Web https://en.wikipedia.org/wiki/Exponential_function, Properties

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